QJEEM23Ma5cc8: JEE Main 2023 Mathematics - Class 12 JEEM23Ma5cc8 Solution: If ∫ − 0.15 0.15

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If $\int_{- 0.15}^{0.15} | 100 x^{2} - 1 | d x = \frac{k}{3000}$ , then $k$ is equal to ___________.

Accepted Answer

\int\limits_{-0.15}^{0.15}\left|100 x^2-1\right| d x=2 \int\limits_0^{0.15}\left|100 x^2-1\right| \mathrm{dx}

\text { Now } 100 x^2-1=0 \Rightarrow x^2=\frac{1}{100} \Rightarrow x=0.1

I=2\left[\int_0^{0.1}\left(1-100 x^2\right) d x+\int_{0.1}^{0.15}\left(100 x^2-1\right) d x\right]

\begin{aligned}I &amp; =2\left[x-\frac{100}{3} x^3\right]_0^{0.1}+2\left[\frac{100 x^3}{3}-x\right]_{0.1}^{0.15} \\\\&amp; =2\left[0.1-\frac{0.1}{3}\right]+2\left[\frac{0.3375}{3}-0.15-\frac{0.1}{3}+0.1\right] \\\\&amp; =2\left[0.2-\frac{0.2}{3}+0.1125-0.15\right] \\\\&amp; =2\left[\frac{5}{100}-\frac{2}{30}+\frac{1125}{10000}\right]=2\left(\frac{1500-2000+3375}{30000}\right) \\\\&amp; =\frac{575}{3000} \Rightarrow \mathrm{k}=575\end{aligned}

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